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Naoki Terai

Publications and source records attributed to Naoki Terai.

At least 19 recordsLinked to original sources
arXiv2026

On the Linearity of Squarefree Powers of Edge Ideals

Let $G$ be a graph and $I(G)$ its edge ideal. The $p$-th squarefree power $I(G)^{[p]}$ is the monomial ideal generated by squarefree monomials corresponding to the matchings of size $p$ of $G$. In this paper, we provide a combinatorial characterization of when $I(G)^{[p]}$ is linearly related, i.e., when its first syzygy module is generated by linear forms. Moreover, for a $1$-dimensional flag simplicial complex $\Delta$ and its Stanley-Reisner ideal $I_{\Delta}$, which arises as the edge ideal of the complement graph of $\Delta$, we describe the shape of the Betti table of $I_{\Delta}^{[p]}$ and we give a combinatorial characterization of when $I_{\Delta}^{[p]}$ has a linear resolution.

math.AC↗
arXiv2025

Algebraic study on rooted products of graphs and multi-clique corona graphs

In this paper, we study rooted products of graphs from the perspective of combinatorial commutative algebra. For edge ideals, we introduce the 2-Cohen-Macaulayness with respect to a vertex and use it to investigate when edge ideals of rooted products of graphs are Cohen-Macaulay. Moreover, we completely determine when attaching a graph on at most six vertices to a given graph as rooted products, yields a Cohen-Macaulay edge ideal. Also, we define mulit-clique corona graphs as a generalization of clique-corona graphs and multi-whisker graphs. We prove that multi-clique corona graphs are vertex decomposable and hence sequentially Cohen-Macaulay. Also, we give formulas for the projective dimension and the Castelnuovo-Mumford regularity.

math.AC↗
arXiv2025

The Serre depth of Stanley-Reisner rings and the depth of their symbolic powers

We investigate an invariant, called the Serre depth, from the perspective of combinatorial commutative algebra. In this paper, we establish several properties of an analogue of the depth of Stanley-Reisner rings. In particular, we relate the Serre depth both to the minimal free resolution of a Stanley-Reisner ring and to that of its Alexander dual. Also, we establish an analogue of a known result that describes the depth of Stanley-Reisner rings in terms of skeletons. Moreover, we study the Serre depth for $(S_{2})$ and the depth on the symbolic powers of Stanley--Reisner ideals. It had been an open question whether the depth of the symbolic powers of Stanley-Reisner ideals satisfies a non-increasing property, but Nguyen and Trung provided a negative answer. We construct an example that the Serre depth for $(S_{2})$ and the depth do not satisfy this property and its second symbolic power is Cohen-Macaulay. Moreover, we prove that the sequence of the Serre depth for $(S_{2})$ on the symbolic powers is convergent and that its limit coincides with the minimum value. Finally, we study the Serre depth on edge and cover ideals. Whether the depth on symbolic powers of edge ideals satisfies a non-increasing property has remained an open question. We address a related problem and show that the Serre depth for $(S_{2})$ on edge ideals of any well-covered graph satisfies a non-increasing property. In addition, we prove that the Serre depth for $(S_{2})$ on the cover ideals of any graph also satisfies a non-increasing property. Moreover, we determine the Serre depth on edge ideals of very well-covered graphs.

math.AC↗
arXiv2025

On minimal free resolutions of the cover ideals of clique-whiskered graphs

We explicitly construct a minimal free resolution of the cover ideals of clique-whiskered graphs. In particular, Cohen--Macaulay chordal graphs, clique corona graphs, and Cohen--Macaulay Cameron--Walker graphs are examples of clique-whiskered graphs. We also introduce multi-clique-whiskered graphs as a generalization of both clique-whiskered graphs and multi-whisker graphs. We prove that multi-clique-whiskered graphs are vertex decomposable and hence sequentially Cohen--Macaulay. Moreover, we provide formulas for the projective dimension and the Castelnuovo--Mumford regularity of their edge ideals. Finally, we construct minimal free resolutions of the cover ideals of both multi-clique-whiskered graphs and very well-covered graphs.

math.AC↗
arXiv2025

The v-numbers of Stanley-Reisner ideals from the viewpoint of Alexander dual complexes

We express the v-number of the Stanley-Reisner ideal in terms of its Alexander dual complex and prove that the v-number of a cover ideal is just two less than the initial degree of the its syzygy module. We give some relation between the v-number of the Stanley-Reisner ideal and the Serre-depth of the quotient ring of the second symbolic power of the Stanley-Reisner ideal of its Alexander dual. We also show that the v-number of the Stanley-Reisner ideal of a 2-pure simplicial complex is equal to the dimension of its Stanley-Reisner ring.

math.AC↗
arXiv2022

Sequentially Cohen-Macaulay binomial edge ideals of closed graphs

In this paper we provide a full combinatorial characterization of sequentially Cohen-Macaulay binomial edge ideals of closed graphs. In addition, we show that a binomial edge ideal of a closed graph is approximately Cohen-Macaulay if and only if it is almost Cohen-Macaulay.

math.AC↗
arXiv2020

Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs

We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.

math.AC↗
arXiv2019

Licci binomial edge ideals

We give a complete characterization of graphs whose binomial edge ideal is licci. An important tool is a new general upper bound for the regularity of binomial edge ideals.

math.AC↗
arXiv2018

Depth and regularity modulo a principal ideal

We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of hypegraphs and of powers of ideals.

math.AC↗
arXiv2012

Cohen--Macaulaynees for symbolic power ideals of edge ideals

Let $S = K[x_1,..., x_n]$ be a polynomial ring over a field $K$. Let $I(G) \subseteq S$ denote the edge ideal of a graph $G$. We show that the $\ell$th symbolic power $I(G)^{(\ell)}$ is a Cohen-Macaulay ideal (i.e., $S/I(G)^{(\ell)}$ is Cohen-Macaulay) for some integer $\ell \ge 3$ if and only if $G$ is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers $I(G)^{(\ell)}$ are Cohen-Macaulay ideals. Similarly, we characterize graphs $G$ for which $S/I(G)^{(\ell)}$ has (FLC). As an application, we show that an edge ideal $I(G)$ is complete intersection provided that $S/I(G)^{\ell}$ is Cohen-Macaulay for some integer $\ell \ge 3$. This strengthens the main theorem in [Effective Cowsik-Nori theorem for edge ideals by M.Crupi, G.Rinaldo, N.Terai, and K.Yoshida, Comm. Alg. 38 (2010), 3347-3357].

math.AC↗
arXiv2012

On the second powers of Stanley-Reisner ideals

In this paper, we study several properties of the second power $I_Δ^2$ of a Stanley-Reisner ideal $I_Δ$ of any dimension. As the main result, we prove that $S/I_Δ$ is Gorenstein whenever $S/I_Δ^2$ is Cohen-Macaulay over any field $K$. Moreover, we give a criterion for the second symbolic power of $I_Δ$ to satisfy $(S_2)$ and to coincide with the ordinary power, respectively. Finally, we provide new examples of Stanley-Reisner ideals whose second powers are Cohen-Macaulay.

math.AC↗
arXiv2010

Cohen-Macaulayness of large powers of Stanley-Reisner ideals

We prove that for m > 2, the m-th symbolic power of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex is a matroid. Similarly, the m-th ordinary power is Cohen-Macaulay for some m > 2 if and only if the complex is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.

math.AC↗
arXiv2010

Vertex decomposability and regularity of very well-covered graphs

A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that $G$ is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the edge ideal of $G$, we show that the Castelnuovo-Mumford regularity of $R/I(G)$ is equal to the maximum number of pairwise 3-disjoint edges of $G$. This improves Kummini's result on unmixed bipartite graphs.

math.AC↗
arXiv2010

Sequentially $S_r$ simplicial complexes and sequentially $S_2$ graphs

We introduce sequentially $S_r$ modules over a commutative graded ring and sequentially $S_r$ simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition $S_r$. In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially $S_r$ if and only if its pure $i$-skeleton is $S_r$ for all $i$. For $r=2$, we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially $S_r$ if and only if the minimal free resolution of the ideal of its Alexander dual is componentwise linear in the first $r$ steps. We apply these results for a graph, i.e., for the simplicial complex of the independent sets of vertices of a graph. We characterize sequentially $S_r$ cycles showing that the only sequentially $S_2$ cycles are odd cycles and, for $r\ge 3$, no cycle is sequentially $S_r$ with the exception of cycles of length 3 and 5. We extend certain known results on sequentially Cohen-Macaulay graphs to the case of sequentially $S_r$ graphs. We prove that a bipartite graph is vertex decomposable if and only if it is sequentially $S_2$. We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially $S_2$. Finally, we propose some questions.

math.AC↗
arXiv2009

Arithmetical rank of lexsegment edge ideals

Let $I\subset S=K[x_1,...,x_n]$ be a lexsegment edge ideal or the Alexander dual of such an ideal. In both cases it turns out that the arithmetical rank of $I$ is equal to the projective dimension of $S/I.$

math.AC↗